Multi-Curve Pricing of Non-Standard Tenor Vanilla Options in QuantLib. Sebastian Schlenkrich QuantLib User Meeting, Düsseldorf, December 1, PDF Free Download

Swaption and caplet pricing w/ Black or Bachelier s formula is trivial. So, why should we care? Banco Popular Espanol SA, Oct. 2027, 3m Euribor +200bp, capped @ 400bp EUR long-term cap prices and vols quoted versus 6m Euribor Non-standard tenor cap and swaption pricing requires consistent and theoretically sound volatility transformation from quoted standard-tenors to required non-standard tenors Multi-Curve Pricing of Non-Standard Tenor Vanilla Options d-fine d-fine All rights All rights reserved reserved 1

Agenda 1. Notation, Normal Volatilities and Tenor Basis Modelling 2. Constant Basis Point Volatility Approach for Swaptions 1. Utilizing CMS pricing results 2. Volatility transformation formula for ATM and skew 3. Caplet Volatility Transformation 1. First Order Approximation of Caplet Dynamics 2. De-correlated ATM Volatility Transformation 3. Skew Transformation 4. Estimating Forward Rate Correlations 1. Deriving 12m Caplet Volatilities from 1y Swaptions 2. Sukzessive Volatility and Correlation Derivation 5. Implementation in QuantLib 6. Summary, Conclusions and Literature Multi-Curve Pricing of Non-Standard Tenor Vanilla Options d-fine d-fine All rights All rights reserved reserved 2

Notation, Normal Volatilities and Tenor Basis Modelling Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Notation, Normal Volatilities and Tenor Basis Modelling d-fine d-fine All rights All rights reserved reserved 3

Volatility transformation methodologies are already discussed in the literature and applied in the industry» Classical approach for log-normal (ATM) volatilities based on Libor (or swap) rates» J. Kienitz, 2013 σ 3m = L6m σ6m L3m ATM Volatility transformation for caplets and swaptions Translate between shifted log-normal and log-normal ATM volatilities to incorporate basis Capture smile by exogenously specifying smile dynamics, e.g. (displaced diffusion) SABR We elaborate an alternative view generalizing available results:» Focus on implied normal volatilities as they become market standard quotations» Consistent basis spread model for swaptions, caps (and exotic derivatives)» Explicitely capture ATM as well as smile by the volatility transformation Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Notation, Normal Volatilities and Tenor Basis Modelling (1/5) d-fine d-fine All rights All rights reserved reserved 4

Affine relation between underlying rates yields model-independent transformation of implied normal volatilities» Implied normal volatilities σ S (K) for an underlying S(T) with expectation S(t) and strike K may be expressed by Bachelier s formula as E S(T) K + = S t K N S t K σ S K T t + N S t K σ S K T t» Consider another underlying U(T) with affine relation U T = a S T + b. Then σ S K T t E U T K + = a E S T K b a + = = U t K N aσ S U t K K b T t a + N U t K K b aσ S T t a aσ S K b a T t» This yields model-independent relation between implied volatilities σ U K = a σ S K b a We derive affine relations between Libor and swap rates and apply above normal volatility transformation Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Notation, Normal Volatilities and Tenor Basis Modelling (2/5) d-fine d-fine All rights All rights reserved reserved 5

Cont. Comp. Rates Tenor basis is modelled as deterministic spread on continuous compounded forward rates f ν (t, T) f μ (t, T) f(t, T) Long tenor curve with forward rates L ν (t; T 1, T 2 ) Short tenor curve with forward rates L μ (t; T 1, T 2 ) OIS discount curve with discount factor P(t, T) Maturity Deterministic spread relation between forward rates f ν t, T = f t, T + b ν (T) f μ t, T = f t, T + b μ (T) Deterministic Relation between forward Libor rates and OIS discount factors 1 + τ 1,2 L ν t; T 1, T 2 = D ν T 1, T 2 P t, T 1 P t, T 2 with D ν T 1, T 2 = e 1 + τ 1,2 L μ t; T 1, T 2 = D μ T 1, T 2 P t, T 1 P t, T 2 with D μ T 1, T 2 = e T2 T1 T2 T1 b ν s ds b μ s ds We use the multiplicative terms D ν and D μ to describe tenor basis Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Notation, Normal Volatilities and Tenor Basis Modelling (3/5) d-fine d-fine All rights All rights reserved reserved 6

Term structure model w/ basis spreads provides benchmark for volatility transformation methodology» Set up Quasi-Gaussaian short rate model (4 factors, local/stochastic volatility, basis spreads)» Compare implied normal volatilities for swaptions and caplets based on 3m and 6m Libor 10y x 10y swaption 5y expiry caplet 6m shifted smile represents 6m smile horizontally shifted by difference in 3m vs. 6m forward rates For a high-level view on volatility transformation just shift the smile by the difference in forwards Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Notation, Normal Volatilities and Tenor Basis Modelling (4/5) d-fine d-fine All rights All rights reserved reserved 7

However, a closer look reveals additional variances in volatilities for different tenors» Same Quasi-Gaussian model and swaption/caplet instruments as before» Zoom in around ATM volatilities What are the reasons for the additional variances?» 3m versus 6m tenor basis» Differences in payment frequency for 3m and 6m Libor legs for swaptions» Interest rate de-correlation for caplets We derive volatility transformation methodologies that capture basis, pay schedules and de-correlation Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Notation, Normal Volatilities and Tenor Basis Modelling (5/5) d-fine d-fine All rights All rights reserved reserved 8

Constant Basis Point Volatility Approach for Swaptions Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Constant Basis Point Volatility Approach for Swaptions d-fine d-fine All rights All rights reserved reserved 9

Forward swap rates are expressed in terms of OIS discount factors and tenor basis spreads» Consider forward swap rates based on short and long tenor Libor rates S μ and S ν with equal annuity» Applying deterministic basis model yields M An T = τ j P(T, T j ) j=1 S μ T = P T, T 0 P T, T M An(T) S ν T = P T, T 0 P T, T M An(T) N μ + 1 An(T) Dμ T μ i 1, T μ μ i 1 P(T, T i 1 ) i=1 N ν + 1 An(T) Dν T ν i 1, T ν ν i 1 P(T, T i 1 ) i=1» Thus S μ and S ν are related by N ν S ν T S μ T = 1 An(T) Dν T ν i 1, T ν ν i 1 P(T, T i 1 ) D μ T μ i 1, T μ μ i 1 P(T, T i 1 ) i=1 N μ i=1 μ/ν Given our basis model, the swap rate spread is stochastic and depends on quotients P(T, T i 1 )/An(T) Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Constant Basis Point Volatility Approach for Swaptions (1/4) d-fine d-fine All rights All rights reserved reserved 10

Terminal Swap Rate (TSR) models developed for CMS pricing provide the ground for swap rate spread modelling» Consider annuity mapping function expressed as linear terminal swap rate model α s, T p = E P(T, T p) An(T) Sμ T = s a T p s S μ t + P(t, T p) An(t) function a T p is derived satisfying additivity and consistency condition with basis spreads for more details on CMS pricing subject to basis spreads, see e.g. S. Schlenkrich, 2015 we may replace terms P(T, T p )/An(T) by a T p S μ (T) S μ t + P t, T p /An(t)» Applying TSR model to swap rate spread yields N ν S ν T S μ T = 1 An t Dν T ν i 1, T ν ν i 1 P(t, T i 1 ) D μ T μ i 1, T μ μ i 1 P(t, T i 1 ) i=1 N ν + S μ T S μ t D ν T ν i 1, T ν ν i 1 a(t i 1 ) D μ T μ i 1, T μ μ i 1 a(t i 1 ) i=1 = S ν t S μ t + S μ T S μ t λ μ,ν N μ i=1 λ μ,ν N μ i=1 Spread component λ μ,ν captures tenor basis and differences in payment schedules Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Constant Basis Point Volatility Approach for Swaptions (2/4) d-fine d-fine All rights All rights reserved reserved 11

Affine relation between swap rates yields implied normal volatility transformation» We have from basis model and Linear TSR model the affine relation S ν T = S μ T + S ν t S μ t + S μ T S μ t λ μ,ν = 1 + λ μ,ν S μ T + S ν t 1 + λ μ,ν S μ t» Combination with the general volatility transformation result yields σ ν K = 1 + λ μ,ν σ μ K Sν t 1 + λ μ,ν S μ t 1 + λ μ,ν» Moreover, for two swap rates S(T) and S(T) based on the same Libor tenor but with different annuities An(T) and An(T) we may derive the following approximate relation S T An t An t S(T) and thus σ K = An t An t An t σ An t K Tenor basis and payment frequency have a slight effect on ATM and skew Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Constant Basis Point Volatility Approach for Swaptions (3/4) d-fine d-fine All rights All rights reserved reserved 12

Swaption transformation methodology is confirmed by benchmark term structure model» Keep 3m curve fixed and mark 3m vs. 6m basis from 0bp to 300bp» Leave remaining parameters (in particular volatilities) unchanged» Analyse resulting implied normal ATM volatilities Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Constant Basis Point Volatility Approach for Swaptions (4/4) d-fine d-fine All rights All rights reserved reserved 13

Basis model yields relation between short and long tenor forward Libor rates» Consider a long tenor period [T 0, T n ] with forward Libor rate L ν t = L ν (t; T 0, T n )» Decomposition into n short tenor periods T 0, T 1,, [T n 1, T n ] with forward Libor rates L i μ t = L i μ t; T i 1, T i» Deterministic spread model yields 1 + τ ν L ν t = D ν T 0, T n P t, T 0 P t, T n 1 + τ i μ L i μ t = D μ T i 1, T i P t, T i 1 P t, T i, i = 1,, n» Fundamental relation between forward Libor rates with 1 + τ ν L ν t = D μ,ν n 1 + τ μ i L μ i t i=1 D μ,ν = D ν (T 0, T n ) D μ (T i 1, T i ) n i=1 T n = exp b ν s b μ (s) ds T 0 Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Caplet Volatility Transformation (1/6) d-fine d-fine All rights All rights reserved reserved 15

First order approximation of Libor rate dynamics» Use vector notation L μ t = L i μ t» Long tenor Libor returns become and define C i=1,,n Lμ n t = 1 + τ μ i L μ i=1 i (t)» The elements of the gradient C L μ t are L ν T L ν t = Dμ,ν τ ν C Lμ T C L μ t Dμ,ν τ ν C Lμ t L μ T L μ t C L μ t = τ μ n j 1 + τ μ i=1 i L i t 1 + τ μ j L μ j (t) j=1,,n = τ j μ 1 + τ ν L ν (t) D μ,ν 1 + τ j μ L j μ (t) j=1,,n» This yields the relation between Libor rates as L ν T L ν t = n i=1 τ i μ 1 + τ ν L ν t τ ν 1 + τ i μ L i μ (t) L i μ T L i μ t v i Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Caplet Volatility Transformation (2/6) d-fine d-fine All rights All rights reserved reserved 16

(Co-)Variance approximation yields relation for ATM volatilities» From L ν T L ν n t = v i L μ i T L μ i=1 i t follows Var L ν T L ν n t = v i v j Cov L μ i T L μ i t, L μ j T L μ j t i,j=1 with v i = τ i μ 1 + τ ν L ν t / τ ν 1 + τ i μ L i t» Approximate variances by implied normal ATM volatilities and correlations as Var L ν T L ν t σ ν L ν t 2 Cov L μ i T L μ i t, L μ j T L μ j t ρ μ i,j σ μ i L μ i t σ μ j L μ j t» This yields ATM volatility transformation formula n σ ν L ν t 2 = v i v j ρ μ i,j σ μ i L μ i t σ μ j L μ j t i,j=1» ATM transformation formula has the same general structure as elaborated in J. Kienitz, 2013 for log-vols. However, v i = τ μ i 1+τ ν L ν t τ ν μ 1+τ i Li t differ and already capture tenor basis Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Caplet Volatility Transformation (3/6) d-fine d-fine All rights All rights reserved reserved 17

Derive smile transformation from boundary argument» Consider special case that» Then L μ (t) = L μ 1 t = = L μ n (t) for all t and thus ρ μ i,j = 1 L ν T L ν t = v i n i=1 L μ T L μ t and we get the affine relation n L ν T = v i i=1 L μ T + L ν t v i n i=1 L μ (t)» This yields the implied volatility smile transformation (for that special case) n σ ν K = v i i=1 σ μ K Lν n t i=1 v i n i=1 v i L μ (t) Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Caplet Volatility Transformation (4/6) d-fine d-fine All rights All rights reserved reserved 18

Combine structure of ATM and (boundary case) smile transformation» ATM volatility transformation n σ ν L ν t 2 = v i v j ρ μ i,j σ μ i L μ i t σ μ j L μ j t i,j=1» Smile transformation for L μ (t) = L 1 μ t = = L n μ (t) n σ ν K = v i i=1 σ μ K Lν n t i=1 v i n i=1 v i L μ (t)» ATM and skew condition represent neccessary conditions for volatility transformation.» We propose the general transformation that complies with ATM and smile condition: σ ν K 2 n = v i v j ρ μ i,j σ μ i K i σ μ j K j i,j=1 with K i = K Lν t n i=1 v i n i=1 v i L i μ t Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Caplet Volatility Transformation (5/6) d-fine d-fine All rights All rights reserved reserved 19

Caplet volatility transformation methodology yields good match with term structure model benchmark» Mark several scenarios for model correlation in Quasi-Gaussian model» Leave remaining parameters unchanged» Analyse model-implied normal ATM volatilities versus model-implied correlations (both evaluated by Monte-Carlo simulation) Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Caplet Volatility Transformation (6/6) d-fine d-fine All rights All rights reserved reserved 20

Consider volatility transformation for caplets as (an invertable) mapping from short tenor volatilities and correlations to long tenor volatilities» Caplet volatility transformation for ATM and smile may be written as ψ: σ μ, ρ μ σ ν» For two long tenor volatilities σ ν 1 and σ ν 2 with μ < ν 1 < ν 2 we also have σ μ ρ μ σν 1 σ ν 2 = ψ1 σ μ, ρ μ ψ 2 σ μ, ρ μ Consequences (provided reasonable regularisation for ρ μ )» If ATM short and long tenor volatilities σ μ and σ ν are available then we can imply correlations ρ μ» If ATM long tenor volatilities σ ν 1 and σ ν 2 are available then we can imply short tenor volatility σ μ and correlation ρ μ Once correlations are derived we can apply caplet volatility transformation for the whole smile Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Estimating Forward Rate Correlation (1/5) d-fine d-fine All rights All rights reserved reserved 22

As a starting point transform 1y ATM swaption volatilities to 12m Euribor caplet volatilities» 1y EUR swaption volatility: 3m Euribor (Act/360) versus semi-annual fixed (30/360) Use swaption transformation methodology to convert 3m Euribor to 12m Euribor ATM vols Use annuity transformation to convert semi-annual, 30/360 to annual Act/360 payments» Transformed volatility is equivalent to 12m Euribor ATM caplet volatility Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Estimating Forward Rate Correlation (2/5) d-fine d-fine All rights All rights reserved reserved 23

Use 6m and 12m caplet volatilities and sukzessively derive further ATM volatilities and correlations Procedure yields consistent setting of volatilities and correlations for all relevant tenors Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Estimating Forward Rate Correlation (3/5) d-fine d-fine All rights All rights reserved reserved 24

We use a classical correlation parametrisation» Correlation structure depending on Libor rate start dates T i and T j ρ i,j = ρ T i + 1 ρ T i exp β(t i ) T j T i, T j > T i» For 3m and 6m correlations mark ρ T i = 0 and only fit β(t i )» For 1m correlation include ρ T i in calibration to improve fit correlation calibration results Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Estimating Forward Rate Correlation (4/5) d-fine d-fine All rights All rights reserved reserved 25

Example calibration of 1m volatilities to (already) available 3m, 6m, and 12m volatilities Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Estimating Forward Rate Correlation (5/5) d-fine d-fine All rights All rights reserved reserved 26

Use decorator design pattern to implement volatility transformation SwaptionVS SmileSection OptionletVS TenorSwaptionVS TenorSwaption- SmileSection TenorOptionlet- SmileSection TenorOptionletVS» Store short/long tenor index» Reference to base volatility» Reference to discount curve» Create TenorSwapttion- SmileSection and pass on volatility evaluation» Set up TSR model» Derive λ μ,ν» Implement volatility transformation» Set up short tenor smile sections» Calculate v i» Implement volatility transformation» Store short/long tenor index» Reference to base volatility» Reference to discount curve» Create TenorOptionlet- SmileSection and pass on volatility evaluation Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Implementation in QuantLib (1/1) d-fine d-fine All rights All rights reserved reserved 28

Summary and Conclusion» Shifting normal volatilities by difference in forward rates provides a reasonable approach for volatility transformation» If more accuracy is desired we elaborated an approach for ATM and smile capturing Tenor basis Payment frequency and day count conventions for swaptions De-correlation for caplets» Combining swaption and caplet volatility transformation allows full specification of correlations and volatilities of non-standard tenors» More details may be found in the literature J. Kienitz. Transforming Volatility - Multi Curve Cap and Swaption Volatilities. 2013. http://ssrn.com/abstract=2204702 M. Henrard. Swaption pricing. Open Gamma Quantitative Research, 2011. S. Schlenkrich, I. Ursachi. Multi-Curve Pricing of Non-Standard Tenor Vanilla Options. 2015. http://ssrn.com/abstract=2695011 S. Schlenkrich. Multi-curve convexity. 2015. http://ssrn.com/abstract=2667405 S. Schlenkrich, A. Miemiec. Choosing the Right Spread. Wilmott, 2015. Multi-Curve Pricing of Non-Standard Tenor Vanilla Options Summary and Conclusion (1/1) d-fine d-fine All rights All rights reserved reserved 30

Dr. Sebastian Schlenkrich Manager Tel +49 89-79086-170 Mobile +49 162-263-1525 E-Mail Sebastian.Schlenkrich@d-fine.de d-fine GmbH Frankfurt München London Wien Zürich Dr. Mark W. Beinker Partner Tel +49 69-90737-305 Mobile +49 151-14819305 E-Mail Mark.Beinker@d-fine.de Zentrale d-fine GmbH Opernplatz 2 D-60313 Frankfurt/Main Tel +49 69-90737-0 Fax +49 69-90737-200 www.d-fine.com d-fine d-fine All rights All rights reserved reserved 31

Multi-Curve Pricing of Non-Standard Tenor Vanilla Options in QuantLib. Sebastian Schlenkrich QuantLib User Meeting, Düsseldorf, December 1, 2015